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  \begin{document}


  
  \vskip 1cm

  \centerline{\Large {Mostly a review}}
\vskip 1cm
  
Most of these are from Lee's book.

\subsection*{Tangent spaces}

\begin{enumerate}[1.]

  \item Let $f:\R^4\to\R^2$ be defined by
    $$f(x,y,s,t)=(x^2+y,x^2+y^2+s^2+t^2+y)$$
    Show that $(0,1)$ is a regular value of $f$ and compute the
    tangent space of $f^{-1}(0,1)$ at $(0,0,0,1)$ as a subspace of
    $\R^4$.



  \item \label{products} If $X,Y$ are manifolds then $X\times Y$ is a
    manifold (you don't have to prove this). Let $(p,q)\in X\times
    Y$. Show that $T_{(p,q)}(X\times Y)$ is the direct sum $T_pX\oplus
    T_qY$ where we identify $T_pX$ with the
    subspace $$Im(d\phi_p:T_pX\to T_{(p,q)}(X\times Y))$$ and
    $\phi:X\to X\times Y$ is inclusion $x\mapsto (x,q)$, and similarly
    for $T_qY$. Hint: It is
    also useful to consider projections $X\times Y\to X$ and $X\times
    Y\to Y$.
\end{enumerate}

\subsection*{Abstract approaches}

  
  
\begin{enumerate}[resume]

    \item For a smooth manifold $X$ we denote by $C^\infty(X)$ the algebra
  of smooth functions $X\to\R$ and by $C^0(X)$ the algebra of
  continuous functions $X\to\R$. Thus $C^\infty(X)\subset C^0(X)$ is a
  subalgebra. When $f:X\to Y$ is a continuous map there is an induced
  ``pull-back'' $f^*:C^0(Y)\to C^0(X)$ defined by $\phi\mapsto \phi
  f$. Show that $f$ is smooth iff $f^*(C^\infty(Y))\subseteq
  C^\infty(X)$.
  
\item For a manifold $X$ and $p\in X$ denote by $\mathcal D_pX$
  the vector space of derivations at $p$. This exercise shows how to
  define the derivative via derivations (thought of as tangent
  vectors) directly without referring
  to charts.

  Let $f:X\to Y$ be a smooth map. Define
    $$df_p:\mathcal D_pX\to \mathcal D_{f(p)}Y$$ by
    $$df_p(D)(g)=D(gf)$$
    (thus for $D\in \mathcal D_pX$ this formula defines $df_p(D)\in
    \mathcal D_{f(p)}$). 
  \begin{enumerate}[(i)]
  \item Verify the chain rule from this definition.
    \item For the case $X=\R^n$ and $Y=\R^m$, $p=0$, $f(p)=0$ and the
      identifications $\mathcal D_0\R^n=\R^n$ from class
      (i.e. $v\in\R^n$ corresponds to the derivation $\partial_v$) and
      likewise for $\R^m$, show that $df_0$ agrees with the usual
      definition of the derivative.
  \end{enumerate}
\end{enumerate}

\subsection*{Lie groups}
\begin{enumerate}[resume]
  
\item Show that Lie groups $O(n)$ and $U(n)$ are compact for $n\geq 1$.

\item Show that Lie groups $GL_n(\R)$ for $n\geq 1$, $SL_n(\R)$ for
  $n\geq 2$, $Sp_{2n}(\R)$ for $n\geq 1$ are
  not compact.




\item Let $G$ be a Lie group, $m:G\times G\to G$ the multiplication,
  and $i:G\to G$ the inversion $g\mapsto g^{-1}$. We will identify
  $T_I(G\times G)$ with $T_I(G)\oplus T_I(G)$ as in Problem \ref{products}.
  \begin{enumerate}[(i)] \item
    Show that the derivative of $m$ at the
    identity $I$ is
    $$dm_I:T_I(G\times G)=T_I(G)\oplus T_I(G), (u,v)\mapsto u+v$$
    Hint: Restrict $m$ to $G\times\{I\}$ and $\{I\}\times G$.
  \item Show that the derivative of $i$ at the identity is
    $$di_I:T_I(G)\to T_I(G), u\mapsto -u$$
    Hint: Consider $G\to G\times G$, $g\mapsto (g,g^{-1})$ and compose
    with $m$.
  \end{enumerate}

  \item Let $G$ be a connected Lie group and $U$ any neighborhood of
    the identity. Show that every $g\in G$ is a product of finitely
    many elements of $U$ (i.e. $U$ generates $G$). Hint: the subgroup
    generated by $G$ is open.

  \item Let $J$ be the diagonal $(n+1)\times (n+1)$ matrix whose
    bottom right entry is $-1$ and all other diagonal entries are
    $1$. The group $O(n,1)$ consists of $(n+1)\times (n+1)$ real
    matrices $M$ such that $MJM^\intercal=J$. This is similar to the
    definition of orthogonal groups, except that instead of preserving
    the standard inner product the matrices must preserve the pairing
    $(x_i)\cdot (y_i)=x_1y_1+\cdots+x_ny_n-x_{n+1}y_{n+1}$. Show that
    $O(n,1)$ is a Lie group and that $$T_I(O(n,1))=\{A\in M(n+1)\mid
    AJ+JA^\intercal=0$$
    The group $O(n,1)$ is essentially (up to some finite index
    subgroups) the group of isometries of the hyperbolic
    $n$-space. The group $O(3,1)$ is the {\it Lorentz group} that
    stars in general relativity.

  \item Show that $O(n,1)$ is not compact, e.g. start with $O(1,1)$.



  \end{enumerate}

      
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